By Gazzola F., Radulescu V.
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Extra resources for A nonsmooth critical point theory approach to some nonlinear elliptic equations in R^n
76) . 78) and v v dP = QT d~ . 80) process v F [~] . B u t =t v the ~t is a the time r e v e r s e d p r o c e s s . 79) we h a v e P = P . Hence under P process, Stochastic diffusion process to d e f i n e a n o t i o n of a c c e l e r a t i o n in s t o c h a s t i c a s e c o n d o r d e r time d e r i v a t i v e . A priori kinematics there exist 48 four b a s i c twice candidates: differentiable D+D+~, D+D_~, as a f u n c t i o n d 2 ~ / d t 2 . H e n c e the m o s t g e n e r a l acceleration on s m o o t h D _ D+~ and D_D_~.
27) order still equal to a model in the past but given the otherwise. in of higher de- This is the Markov ~E[d~ t d~ t t ~t ] = ~iJ(~t,t)dt with the conditional t+dt that the process of the process for a process and In other words, are independent. 28) Brownian motion, wio = o [dW t I ~t = x] = 0 [dW~ dW~ or, in integral I ~t = x] = ~iJdt form t i x i ~t (s,x) = + ~ b±(~T(s,x),T)dT t n ~ , + j_Z_1 ! 29) integral is taken in the sense of It8 . 30) and can easily be computed the following (2.
If some external force is present), we can no longer expect to have ther ~ [d~tl~ t] = f(~t,t,dt) If we drop ~ [d~t[~t] = O , but ra- for some function f , if this expecta- tion exists. 26) ]E [d~tl~ t] = b ( ~ t , t ) d t + o(dt) This a s s u m p t i o n is both intuitively reasonable and t e c h n i c a l l y powerful. Next, note that for the W i e n e r process all the random variables in the m o r e general situation. if d~ t function is i n d e p e n d e n t of b(x,t) dw t is independent of Ws,S < t , but we cannot expect this p r o p e r t y Indeed, ~t ' then in the situation discusses above, ~ (d~tl~t) = ~ (d~t) , and the w o u l d have to be spacially constant.
A nonsmooth critical point theory approach to some nonlinear elliptic equations in R^n by Gazzola F., Radulescu V.